Question

Write each of the relations below in algebra and see if it gives a polynomial. Also, give reasons for your conclusion.

i. A 1-metre wide path goes around a square ground. The relation between the length of a side of the ground and the area of the path.

ii A liquid contains 7 litres of water and 3 litres of acid. More acid is added to it. The relation between the amount of acid added and the change in the percentage of acid in the liquid.

iii. Two poles of heights 3 metres and 4metres are erected upright on the ground, 5 metres apart. A rope is to be stretched from the top of one pole to some point on the ground and from there to the top of the other pole:

The relation between the distance of the point on the ground from the foot of a pole and the total length of the rope.

Answer 1

i. Let the side of the ground be x then the side of the square including the path is x + 2 metres.

Area of path = Area of large square – Area of small square.

a(x) = (x + 2)² – x² = x² + 4x + 4 – x²

= 4x + 4

Here a(x) is a polynomial. Here x is multiplied by 4 and 4 is added to it.

ii. Ratio of acid in the first fluid = 3/10 litre that is 30%

If x litres of acid is added to it, then change in the amount of acid is

\(=\frac{3+x}{10+x}\)

Change in the percentage of acid is = \(\frac{3+x}{10+x}\times100\%\)

\(b(\text{x})=\frac{300+100x}{10+x}\%\)

iii.

In figure, AB = x m, BD = 5 – x 

Total length of the wire = BC + BE

It is not a polynomial since it involves square root of variable x.